Consider a spacetime $(\zeta^{3,1},g)$

where $$g=\frac{du\,dv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2} \quad \forall u,v,r,w \in (0,1)$$ Now this is just Minkowski space in different coordinates (related to Dirac/Light cone coordinates/null coordinates). I'm looking to take a Cauchy foliation of $\zeta^{3,1},$ change the metric back to $g'=dudv-dw^2-dr^2,$ and use the induced measure from $g'$ by means of the volume form to transform the foliated past light cone region onto a new spacetime $(\Psi^{~3,1},d).$

Let's look at the steps involved for the $(1+1)$ dimensional case $(\zeta^{1,1},u)$ where $u=\frac{dxdq}{xq}.$ The Cauchy foliation is simply $\ln(b)\ln(y)=t$ (solving for $y$ gives an explicit representation) which can then be transformed via Mellin-like transform:

$$\Phi_s(t)= \int_{S=(0,1)} \exp {\frac{t s}{\ln b}}~db = \int_0^1 tb^{t-1} \exp \frac{s}{\ln b}~db = 2\sqrt{ts}K_1(2\sqrt{ts})$$

Observe that this is an unnormalized K-distribution and it's inverse transform yields an unnormalized distribution call it the "$\zeta$-distribution" (our Cauchy foliation). Observe that the Fisher information metric of the $\zeta$-distribution is $\Phi_s(t)$ up to a factor. Observe that the $\zeta$-distribution yields solutions to the Killing field $X=\langle x \ln x, -y\ln y \rangle.$ It also provides a particular distributional solution to the following diffusion equation with diffusivity depending on both space and time:

$$\frac{\partial^2}{\partial t^2}\nu(t,b)=-\frac{b}{t}\frac{\partial}{\partial b}\nu(t,b)$$

In short, $\Phi_1(t)=v$ is a Lorentzian metric on the smooth $(1+0)$ manifold $(\Psi^{~1,0}, v).$ We have that $\Psi^{~3,1}=\Psi^{~1,1}\times \Psi^{~1,0} \times \Psi^{~1,0}$ so we can define the product metric on $\Psi^{~3,1}.$

There are two different constructions going on here. The first construction starts with a bonafide spacetime and transforms the past light cone region onto a new spacetime. The second construction begins by simplifying the spacetime down to $(1+1)$ dimensions, transforms the past light cone region, and then scales that manifold with the Cartesian product, furnishing with the product metric.

Does restricting to $g'$ and transforming the foliated past light cone preserve all or most of the information contained in $(\zeta^{3,1},g)?$ Is $(\Psi^{~3,1},d)$ also a spacetime, or a component of one?